| L(s) = 1 | − 2-s + 4-s + 5-s − 5·7-s − 8-s − 10-s + 5·11-s − 13-s + 5·14-s + 16-s + 5·17-s − 3·19-s + 20-s − 5·22-s − 3·23-s + 25-s + 26-s − 5·28-s − 6·29-s − 6·31-s − 32-s − 5·34-s − 5·35-s − 37-s + 3·38-s − 40-s + 4·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.88·7-s − 0.353·8-s − 0.316·10-s + 1.50·11-s − 0.277·13-s + 1.33·14-s + 1/4·16-s + 1.21·17-s − 0.688·19-s + 0.223·20-s − 1.06·22-s − 0.625·23-s + 1/5·25-s + 0.196·26-s − 0.944·28-s − 1.11·29-s − 1.07·31-s − 0.176·32-s − 0.857·34-s − 0.845·35-s − 0.164·37-s + 0.486·38-s − 0.158·40-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.506167921955232129928080315022, −7.26238630113822102791797748853, −6.93245784370521961612621544079, −6.00025851812158853517523212456, −5.70716897529375252249064677689, −4.02906718830663741743009133501, −3.47738567939455373126598827256, −2.47043245971722245325321389426, −1.33411491837942943308142579279, 0,
1.33411491837942943308142579279, 2.47043245971722245325321389426, 3.47738567939455373126598827256, 4.02906718830663741743009133501, 5.70716897529375252249064677689, 6.00025851812158853517523212456, 6.93245784370521961612621544079, 7.26238630113822102791797748853, 8.506167921955232129928080315022
